Regularity for a singular conservation law

A Ow Conservation Law for Surface Processes

The object studied in this paper is a pair ((; Y), where is a random surface in I R N and Y random vector eld on I R N. The pair is jointly stationary, i.e. its distribution is invariant under translations. The vector eld Y is smooth outside but may have discontinuities on. Gauss' divergence theorem is applied to derive a ow conservation law for Y. For I R 1 this specializes to a well-known rat...

Regularity through Approximation for Scalar Conservation Laws∗

In this paper it is shown that recent approximation results for scalar conservation laws in one space dimension imply that solutions of these equations with smooth, convex fluxes have more regularity than previously believed. Regularity is measured in spaces determined by quasinorms related to the solution’s approximation properties in L1(R) by discontinuous, piecewise linear functions. Using a...

High Order Regularity for Conservation Laws

We study the regularity of discontinuous entropy solutions to scalar hyperbolic conservation laws with uniformly convex fluxes posed as initial value problems on R. For positive α we show that if the initial data has bounded variation and the flux is smooth enough then the solution u( · , t) is in the Besov space Bα σ (L σ) where σ = 1/(α + 1) whenever the initial data is in this space. As a co...

Well-posedness of a singular balance law

We define entropy weak solutions and establish well-posedness for the Cauchy problem for the formal equation ∂tu(t, x) + ∂x u 2 (t, x) = −λu(t, x) δ0(x), which can be seen as two Burgers equations coupled in a non-conservative way through the interface located at x = 0. This problem appears as an important auxiliary step in the theoretical and numerical study of the one-dimensional particle-in-...

The Conservation Law